Integral Projection Models

This chapter covers a more advanced topic in (structured) population modelling, integral projection models (IPM). We will first derive the formula for IPM analogously to what we learned in Chapter 2, matrix projection models (MPM). Next, we will introduce the method to build simple IPM with case studies. We will then move on to the analysis of IPM, including asymptotic behaviors, sensitivity and elasticity. Lastly, we will incorporate density-dependence and stochasticity into our IPM, just as what we did in previous chapters.

Learning outcomes

Understand the analogy between IPM and MPM, and the mathematics behind IPM
Being able to choose the proper model for different species based on their life cycles, and construct simple IPM from data
Getting familiar with various numerical methods, such as regression, numerical integration
Being able to perform basic analysis of IPM

Introduction: MPM vs. IPM

Matrix projection models (MPM) are widely used in modelling structured biological populations. They are easy to understand and are conceptually simple to represent population structures. A matrix model divides the population into a set of classes, which give rise to some potential problems (Ellner and Rees, 2006):

Representing individuals with different characteristics as a distinct stage or category is bound to introduce some amount of inaccuracy
Increasing the number of stages leads to higher sampling error because of fewer available data

In fact, for some organisms, instead of dividing their life cycles into discrete classes (e.g. juveniles and adults), it is more appropriate to use continuous variables (e.g. body size) (Ellner and Rees, 2006). Easterling et al. (2000) proposed integral projection models (IPM) as an alternative to MPM for populations in which demographic rates are primarily influenced by a continuous variable. This kind of model was analogous to MPM, allowing researchers to avoid inaccurate classifications.

Contruction of IPM

In this section, we will introduce the basic concepts underlying IPM. We will make the assumption that differences between individuals in a population are completely described by a continuous variable \(z\), which could be some continuous measure of body size (e.g. total mass and volume). \(z\) could also be unrelated to “size”, such as the individual’s spatial location in a linear habitat. However, \(z\) must have finite limits.

Model structure

Let \(n(z,t)\) be the size distribution of individuals at time \(t\). The number of individuals with size \(z\) in the interval \([a,b]\) is \[\int_a^bn(z,t)\ dz.\]

This may seem a bit hard to follow. Here we present a more intuitive description:

The number of individuals in the size interval \([a,b]\) at time \(t\) is approximately \(n(a,t)(b-a)\), given a very small difference \(b-a\).

This comes directly from the intuition behind an integral, which we will show in the following figures:

Illustration of an integral with its rectangular approximation

The integral \(\int_a^bn(z,t)\ dz\) is equal to the area under the curve, which is shaded in red (stripe); \(n(a,t)(b-a)\) is equal to the rectangular area shaded in red (stripe) and blue (solid). When \(b\) is close enough to \(a\), the blue area is small enough to be ignored, making \[\int_a^bn(z,t)\ dz\approx n(a,t)(b-a).\]

Now if we assume the limit of \(z\) is from \(L\) to \(U\) (i.e. \(z\in[L,U]\)), the total population is the integral of \(n(z,t)\) over the domain \([L,U]\), \[N(t)=\int_L^Un(z,t)\ dz.\] We should notice that \(n(z,t)\) is different from a probability distribution, whose integral over the domain is 1. Another thing we should notice is that \(n(z,t)\) is NOT the number of individuals of size \(z\) at time \(t\). If we want to know the number of individuals in a size range, we have to calculate the integral.

Kernel

Similar to MPM, IPM also operates in discrete time. From \(t\) to \(t+1\), individuals could die or change in size. They could also produce offsprings that vary in size. We use two functions, \(P(z',z)\) and \(F(z',z)\), to describe these two types of size transitions (and for convenience, we further let \(z'\) be the size at time \(t+1\)). The total transition is represented by \(K(z',z)=P(z',z)+F(z',z)\), which is called the kernel.

Survival/growth kernel

\(P(z',z)\) is called the survival/growth kernel (Merow et al., 2014), which is often written as \(P(z',z)=s(z)G(z',z)\), where \(s(z)\) is the survival rate and \(G(z',z)\) represents the size transition. We can think of \(G(z',z)\) as the probability density function of the subsequent size \(z'\) of an size-\(z\) individual, so we always have \[\int_L^UG(z',z)\ dz'=1.\] The idea can be explained with the following plot

Illustration of size transitions and \(G(z',z)\)

Suppose that we have collected data of sizes at \(t\) and \(t+1\), we assume that, for every \(z\), the subsequent size \(z'\) follows some probability distribution, which is represented by the red curves.

Fecundity kernel

\(F(z',z)\) is called the fecundity kernel (Merow et al., 2014), which represents reproduction of offsprings. \(F(z',z)\) is analogous to a probability density function. It is the size distribution of offsprings produced by a size-\(z\) individual. In other words, the number of the offsprings of size \(z'\) in the interval \([a,b]\) by a size-\(z\) individual is \[\int_a^bF(z',z)\ dz'.\] That is, it follows a similar idea of \(n(z,t)\).

Now, putting everything together, the population size distribution at time \(t+1\) can be calculated as the integral \[n(z',t+1)=\int_L^UK(z',z)n(z,t)\ dz.\] The kernel \(K(z',z)\) in the IPM is analogous to the projection matrix in MPM.

From life cycle to model

An essential step of building an IPM is to translate population census data into the vital rates in the kernel.

Population census

First, we introduce two different census methods:

Pre-reproductive census: Census takes place before reproduction. This is common in plants, as the size of offsprings are hard to collect before growth to noticeable small individuals.

Post-reproductive census: Census takes place after reproduction. This is more common in animals, when we can easily distinguish newborns from adults and measure their sizes.

Building kernels

Once we decided the census method, and collected data at each time point, it’s time to build the kernel for our IPM, which will be divided into two parts:

Survival: After we observed the number of individuals survived from each census to the next, we can estimate the survival rate \(s\). By recording the size changes, we can also obtain the distribution of size transition \(G\).
Reproduction: We can estimate some basic parameters: per capita offspring number \(b\), and probability of successful recruitment \(p_r\). When we are able to distinguish breeders and non-breeders, we can further define the probability of reproducing \(p_b\). Similarly, by recording the recruit sizes, we can also obtain the size distribution of recruits, \(C\).

For different census methods, kernels are defined differently. Here we present the most basic situations for both census methods.

	Pre-reproductive census	Post-reproductive census
Survival kernel	\(s(z)G(z',z)\)	\(s(z)G(z',z)\)
Fecundity kernel	\(p_b(z)b(z)p_rC(z',z)\)	\(s(z)p_b(z)b(z)C(z',z)\)

The survival kernel is pretty much the same, while the fecundity kernel is slightly different between two census methods. This is because

In a post-reproductive census, new recruits are censused before mortality, so every new recruit is essentially “successful”, making \(p_r=1\) in the fecundity kernel.
In a pre-reproductive census, since the census happens right before reproduction, it is usually assumed that no death occurs between census and reproduction, making \(s(z)=1\) in the fecundity kernel.

Now, let’s first look at the total population at the time \(t+1\), \(N(t+1)\). It can be calculated as \[N(t+1)=\int_L^U\underbrace{[s(z)+p_b(z)b(z)p_r]}_{\text{kernel}}n(z,t)\ dz\] in a pre-reproductive census. Considering the size transition, we further have \[n(z',t+1)=\int_L^U\underbrace{[s(z)G(z',z)+p_b(z)b(z)p_rC(z',z)]}_{\text{kernel}}n(z,t)\ dz.\]

Similarly, in a post-reproductive census, we have \[n(z',t+1)=\int_L^U\underbrace{[s(z)G(z',z)+s(z)p_b(z)b(z)C(z',z)]}_{\text{kernel}}n(z,t)\ dz.\]

Note: based on different census methods, the recruit size distribution \(C(z',z)\) will be different. For pre-reproductive census, \(C(z',z)\) is the size distribution of new recruits at age 1 (after they grow to noticeable individuals after one year), so we denote it as \(C_1(z',z)\); for post-reproductive census, \(C(z',z)\) is the size distribution of new recruits at age 0 (immediately after they are born), so we denote it as \(C_0(z',z)\).

Vital rates with examples

Here, we present some examples of kernels based on life cycles of different species.

Example 1: long-lived perennial plant with no seed bank

This example comes from Merow et al. (2014) and it represents the most basic case. For plants, key life history transitions usually depend more on size than on age (Ellner et al., 2016). We assume that once seeds germinate, individuals grow until they are large enough to produce seeds, after which they continue to reproduce until they die. The survival/growth kernel is \[P(z',z)=s(z)G(z',z),\] and the fecundity kernel is \[F(z',z)=p_b(z)b(z)p_rC_1(z').\] The kernel is just the sum \[K(z',z)=s(z)G(z',z)+p_b(z)b(z)p_rC_1(z').\] We should notice that

\(p_r\) describes the ratio of recruits observed at \(t+1\) compared to the seeds produced at \(t\) (Metcalf et al., 2008).
\(C_1(z')\) is size distribution of recruits observed at time \(t+1\). It does not depend on \(z\), since we do not know the parental size.

Example 2: monocarpic perennial

The following two examples were discussed in Ellner et al. (2016). We will also use them as case studies in the next section. This example is based on Oenothera glazioviana, a monocarpic plant that often occurs in sand dune areas. For a monocarpic plant, we should keep in mind that reproduction is fatal. The kernel function we will study is \[K(z',z)=(1-p_b(z))s(z)G(z',z)+p_b(z)b(z)p_rC_1(z').\] We should notice that

The factor \((1-p_b(z))\) in the survival/growth kernel represents the proportion of non-flowering plants, as flowering plants die after reproduction.
Similar to Example 1, since we are unable to track parentage, the recruit size distribution is independent of parental size \(z\).

© Bernd Sauerwein, Oenothera glazioviana, Creative Commons CC BY-SA-3.0 license

Example 3: ungulate

Let’s look at a classic animal example, in which we are able to track parental sizes. For animals, age or life stage (e.g. mature vs. immature) plays a more important role in life history transitions. Here we present a example of Soay sheep (Ovis aries), whose demographic rates are assumed to be functions of body mass. The kernel function is \[K(z',z)=s(z)G(z',z)+s(z)p_b(z)p_r\left(\frac{C_0(z',z)}{2}\right).\] We should notice that

We are using post-reproduction census, so the survival rate \(s(z)\) appears in both survival/growth and fecundity kernels.
We assume that a single lamb is born at each reproductive event, so \(b(z)=1\).
The factor \(1/2\) appears in the fecundity kernel because we only track female population and an equal sex ratio is assumed.
Since we know the parents of each newborn, the recruit size distribution should be dependent on both \(z\) and \(z'\).

Illustration of current application: spatial integral projection model

As we mentioned earlier, the continuous variable \(z\) could be unrelated to “size”. Also, \(z\) could be multidimensional (Ellner et al., 2016). Here we present a paper about spatial integral projection models (SIPM) that include both demography and dispersal with continuous variables.

Jongejans, E., Shea, K., Skarpaas, O., Kelly, D. and Ellner, S.P. (2011), Importance of individual and environmental variation for invasive species spread: a spatial integral projection model. Ecology, 92: 86-97.

© Baykedevries, Carduus nutans, Creative Commons CC BY-SA-3.0 license

Snapshot of the study

The authors developed an SIPM by combining IPM with an analytical method for modelling spatial spread, Neubert-Caswell Model (NCM). They tested the SIPM on the invasive monocarpic thistle Carduus nutans in New Zealand.
For the demographic process, the population is described by the size distribution of rosettes, \(R(z,t)\), \[R(z',t+1)=\int_Z K_{dem}(z',z)R(z,t)\ dz.\]
The vital rates underlie the demographic kernel \(K_{dem}(z',z)\) are \[K_{dem}=\underbrace{s(z)(1-p_b(z))G(z',z)}_{\text{survival/growth kernel }P(z',z)}+\underbrace{s(z)p_b(z)b(z)p_r\omega(z)\phi C_1(z')}_{\text{fecundity kernel }F(z',z)}.\] Compared to the above example of monocarpic perennials, the kernel is defined differently;
- Survival rate also appears in the fecundity kernel
- Two more variables are introduced. \(\omega(z)\) is a size-dependent potential flower head production term, and \(\phi\) is the probability that a potential seed escapes florivory and becomes a seed.
A seed bank is also considered in the model. Basically, the existence of a seed bank requires the addition of a separate discrete-state variable that is linked to \(K_{dem}\) by the proportion of newly produced seeds that enter the seed bank, and the proportion of seeds in the seed bank that survive and become seedlings. We will not discuss the details, but Merow et al. (2014) provides some examples of plants with seed banks.
Spatial population spread requires a dispersal kernel \(K_{disp}(r)\), which is the probability density function for seed dispersal distance \(r\). The authors used an inverse Gaussian distribution.
The SIPM that combines both demographic kernel and dispersal kernel is described as a double integral \[\mathbf{n}(x',z',t+1)=\iint K(x',z',x,z)\mathbf{n}(x,z,t)\ dxdz,\] where \(\mathbf{n}(x,z,t)\) is the structured population stage vectors at location \(x\) with size \(z\) at time \(t\). \(K(x',z',x,z)\) is the combined demography and dispersal kernel, which is defined as follows \[K(x',z',x,z)=K_{disp}(x'-x)F(z',z)+\delta(x'-x)P(z',z),\] with \(\delta(\cdot)\) the Dirac delta function (\(\delta(x'-x)=0\) for all \(x'\neq x\) and \(\delta(x'-x)=1\) for \(x'=x\))
- Note: Only new seeds disperse, while all other life stages are sessile.

Reference

Easterling, M.R., Ellner, S.P. and Dixon, P.M. (2000), Size-specific sensitivity: applying a new structured population model. Ecology, 81: 694-708.

Ellner, S.P. and Rees, M. (2006), Integral projection models for species with complex demography. Am Nat., 167(3): 410-28.

Ellner, S.P., Childs, D.Z. and Rees, M. (2016), Data-driven Modelling of Structured Populations. Springer Cham.

Merow, C., Dahlgren, J.P., Metcalf, C.J.E., Childs, D.Z., Evans, M.E.K., Jongejans, E., Record, S., Rees, M., Salguero-Gómez, R. and McMahon, S.M. (2014), Advancing population ecology with integral projection models: a practical guide. Methods Ecol Evol, 5: 99-110.

Metcalf, C., Rose, K., Childs, D., Sheppard, A., Grubb, P. and Rees, M. (2008), Evolution of flowering decisions in a stochastic, density-dependent environment. Proceedings of the National Academy of Sciences, 105: 10466.